Question

To find the two incomparable elements in the poset \((\{ 1,2,4,6,8\} ,1)\).

Short answer

The two elements \((0,1)\) are incomparable elements of the poset \((\{ 1,2,4,6,8\} ,\mid )\).

Step-by-step solution

Given data

The poset \((\{ 1,2,4,6,8\} ,\mid )\).

Concept used of partially ordered set

A relation\(R\)is a poset if and only if,\((x,x)\)is in\({\rm{R}}\)for all x (reflexivity)

\((x,y)\)and\((y,x)\)in R implies\(x = y\)(anti-symmetry),\((x,y)\)and\((y,z)\)in R implies\((x,z)\)is in\({\rm{R}}\)(transitivity).

Compare the pair of elements

Consider the elements of a poset \((P, \le )\) are \(a,b\). If \(a \le b,b \le a\) then the elements are said to be comparable.

Now consider the poset \((\{ 1,2,4,6,8\} ,\mid )\).

From given poset, we observe that \(4not\mid 6{\rm{ ,}}6\) not| 4

Then the elements \((4,6)\) are two incomparable elements of the poset \((\{ 1,2,4,6,8\} ,1)\).